THE ALEKSANDROV PROBLEM AND THE MAZUR-ULAM THEOREM ON LINEAR n-NORMED SPACES

Abstract

Abstract. This paper generalizes the Aleksandrov problem and MazurUlam theorem to the case of n-normed spaces. For real n-normed spacesX and Y, we will prove that f is an affine isometry when the mappingsatisfies the weaker assumptions that preserves unit distance, n-colinearand 2-colinear on same-order. 1. IntroductionLet X and Y be metric spaces. A mapping f : X → Y is called an isometryif f satisfies d Y (f(x),f(y)) = d X (x,y) for all x,y ∈ X, where d X (·,·) andd Y (·,·) denote the metrics in the spaces X and Y , respectively. For some fixednumber r > 0, assume that f preserves the distance r, i.e., for all x,y ∈ X withd X (x,y) = r, it holds that d Y (f(x),f(y)) = r. Then r is called a conservative(or preserved) distance for the mapping f.In 1970, Aleksandrov [1] posed the following problem: Examine whether theexistence of a single conservative distance for some mapping T implies that Tis an isometry.In 1932, Mazur and Ulam [8] proved a theorem that every isometry of a realnormed space onto a real normed space is a linear mapping up to a translation.In 1993, Rassias and Semrl [10] proved a series of results on the DOPPˇ(distance one preserving property) for the normed spaces (see also [6, 7, 9]).Since 2004, the Aleksandrovproblem has been discussedin the n-normed spaces(n ≥ 2) (see [2, 3, 4, 5]). Chu, Lee and Park proved:Theorem 1.1 ([3]). Let X and Y be two real linear n-normed spaces. If amapping f : X → Y satisfies the following conditions:(1) f has the n-DOPP;(2) f is n-Lipschitz;